Pathwise uniqueness for singular SDEs driven by stable processes

Abstract

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable Lévy processes with values in $\mathbb{R}^{d}$ having a bounded and $\beta$-Hölder continuous drift term. We assume $\beta > 1 - \alpha/2$ and $\alpha \in [1, 2)$. The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.